In this context, I would take any list of rules that dictates buying securities to be an index.

For example, a fund buys an equal number of shares of every publicly listed company.

Or a fund buys securities that trade with a ticker symbol starting with the letter C.

Based on that definition, one could refer to a Cathie Wood index fund to be composed of whatever she decides to buy and a bagacrap index fund to be composed of whatever bagacrap chooses to buy.

And if you went with that maximalist definition of an index, then the word would have no useful meaning. If the inclusion criteria is completely arbitrary and capricious (i.e. whatever cathie woods decides she likes that week), then it's still a fund, it's just not an index fund.

QQQ is both arbitrary (the stock exchange has nothing to do with business operations) and capricious (they are changing the rules on a whim to serve Elon's interests, and because the arbitrary inclusion criteria somewhat forces them to).